Self-similar solutions are obtained for the supersonic compression of a plasma containing an axial current and an entrained axial magnetic field (screw-pinch). These solutions represent a complete generalization of previous works done on the Z-pinch and $\Theta$-pinch schemes. The solutions presented herein represent a screw-pinch plasma with a diffuse current profile. Four separate plasma implosion modes are identified. Apart from constituting a mathematical tour-de-force, the solutions representing the implosion modes differ qualitatively from the Z-pinch and $\Theta$-pinch schemes.They comprise of the plasma annulus implosion, a converging transverse MHD shock, the propagation of fast magnetosonic waves towards the axis (weak discontinuities), and, finally, the collapse of a hollow plasma liner. These modes are chosen on the basis of their ability to achieve magnetic field cumulation during the course of the implosion. Analytic expressions describing the asymptotic behavior of the hydromagnetic profiles are obtained for each of the four implosion modes.Numerical examples for selected boundary conditions, representing realistic plasma parameters for the purpose of ultra-high magnetic field generation, are presented for each of the four implosion modes.A special case representing the exponential implosion of a screw-pinch plasma has also been analyzed. It has been found that specifying an exponential dependence to the temporal contribution of the hydromagnetic variables, results in a more constrained solution space. This comparison is made with the case described above which has a power law type temporal contribution. It has been found that the special case admitted only the plasma annulus implosion mode. A few numerical examples for this mode have been provided.Owing to the immense possible applications associated with plasma pinches, the time independent profiles of the hydromagnetic variables have been utilized in order to arrive at an estimate for the practical utility of the theoretical model. More specifically, the time dependent profiles of the hydromagnetic variables, and a piston condition for an external laser driver have been obtained. Numerical examples have also been provided for two cases representing screw-pinch plasma implosions occurring for realistic experimental parameters.